# American Institute of Mathematical Sciences

June  2021, 20(6): 2323-2340. doi: 10.3934/cpaa.2021081

## On local solvability for a class of generalized Mizohata equations

 1 Universidade Federal do Rio Grande, Instituto de Matemática, Estatística e Física, RS, Brazil 2 Universidade Federal de São Carlos, Departamento de Matemáica, SP, Brazil

* Corresponding author

Received  October 2020 Revised  March 2021 Published  June 2021 Early access  May 2021

Fund Project: The first author was partially supported by CAPES. The second author is a associate researcher of the Thematic project no. 2018/14316-3 (FAPESP)

The image in $C^{\infty}$ for a class of complex vector fields, containing the Mizohata operator, was characterized.

Citation: L. R. Nunes, J. R. Dos Santos Filho. On local solvability for a class of generalized Mizohata equations. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2323-2340. doi: 10.3934/cpaa.2021081
##### References:

show all references

##### References:
Integration domain complexification
 [1] Patrick Bonckaert, P. De Maesschalck. Gevrey and analytic local models for families of vector fields. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 377-400. doi: 10.3934/dcdsb.2008.10.377 [2] Onur Alp İlhan. Solvability of some partial integral equations in Hilbert space. Communications on Pure & Applied Analysis, 2008, 7 (4) : 837-844. doi: 10.3934/cpaa.2008.7.837 [3] Wei Xi Li, Chao Jiang Xu. Subellipticity of some complex vector fields related to the Witten Laplacian. Communications on Pure & Applied Analysis, 2021, 20 (7&8) : 2709-2724. doi: 10.3934/cpaa.2021047 [4] Vincent Naudot, Jiazhong Yang. Finite smooth normal forms and integrability of local families of vector fields. Discrete & Continuous Dynamical Systems - S, 2010, 3 (4) : 667-682. doi: 10.3934/dcdss.2010.3.667 [5] Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934/proc.2015.0754 [6] Leonardo Câmara, Bruno Scárdua. On the integrability of holomorphic vector fields. Discrete & Continuous Dynamical Systems, 2009, 25 (2) : 481-493. doi: 10.3934/dcds.2009.25.481 [7] Jifeng Chu, Zhaosheng Feng, Ming Li. Periodic shadowing of vector fields. Discrete & Continuous Dynamical Systems, 2016, 36 (7) : 3623-3638. doi: 10.3934/dcds.2016.36.3623 [8] Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481 [9] Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264 [10] Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703 [11] Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053 [12] Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515 [13] Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032 [14] Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167 [15] Paul Bracken. Exterior differential systems and prolongations for three important nonlinear partial differential equations. Communications on Pure & Applied Analysis, 2011, 10 (5) : 1345-1360. doi: 10.3934/cpaa.2011.10.1345 [16] Marcus A. Khuri. On the local solvability of Darboux's equation. Conference Publications, 2009, 2009 (Special) : 451-456. doi: 10.3934/proc.2009.2009.451 [17] José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213 [18] BronisŁaw Jakubczyk, Wojciech Kryński. Vector fields with distributions and invariants of ODEs. Journal of Geometric Mechanics, 2013, 5 (1) : 85-129. doi: 10.3934/jgm.2013.5.85 [19] Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021, 41 (10) : 4943-4957. doi: 10.3934/dcds.2021063 [20] A. V. Bobylev, Vladimir Dorodnitsyn. Symmetries of evolution equations with non-local operators and applications to the Boltzmann equation. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 35-57. doi: 10.3934/dcds.2009.24.35

2020 Impact Factor: 1.916